Powers of Negative Numbers: Adding parentheses in the correct place

To solve this problem, we need to evaluate expressions by applying the rules of exponents and the effects of parentheses on negative numbers:

• $(-3)^2$: When a negative number is squared, the result is positive. So, $(-3)^2$ means $-3 \times -3 = 9$.
• $-(-3)^2$: This means $-1 \times (-3 \times -3)$ because squaring a number negates the negative sign inside parentheses, resulting in $-9$.
• $-(3)^2$: This equals $-1 \times (3 \times 3) = -9$, as the negative sign is outside the squared value.
• $-3$: This is simply $-3$.

Only $(-3)^2$ equals 9, confirming it as the correct expression required by the problem.

Therefore, the solution to the problem is $(-3)^2$.

To solve this problem and express 64 as a power involving a negative number, we will follow these steps:

• Step 1: Recognize that we need to represent 64 using a base number squared. Since 64 is a perfect square, let's consider negative integers whose square equals 64.
• Step 2: The principal positive square root of 64 is 8. However, we are tasked with finding a negative number such that its square is 64.
• Step 3: If we have a negative integer, $(-8)$, and square it, we have: $(-8)^2 = (-8) \times (-8) = 64$.
• Step 4: Compare this with the expression $-(8)^2$, which results in $-64$ because the square applies only to 8, and the negative sign flips the result.

Therefore, the correct expression representing 64 with a negative base is $(-8)^2$, and among the answer choices provided, choice 1 is the correct one.

To solve this problem, let's evaluate both given expressions to determine which results in 8.

• Step 1: Evaluate $(-2)^3$:
  $(-2)^3 = (-2) \times (-2) \times (-2)$.
  Multiplying across: $(-2) \times (-2) = 4$, and then $4 \times (-2) = -8$.
  Thus, $(-2)^3 = -8$.
• Step 2: Evaluate $-(-2)^3$:
  First, calculate $(-2)^3$ again: We already know $(-2)^3 = -8$.
  Now, apply the negative sign: $-(-8) = 8$.

Therefore, the expression that equals $8$ is $-(-2)^3$.

Thus, the correct expression that evaluates to 8 is $-(-2)^3$.

To determine which expression equals 36, we need to consider how squaring works with negative numbers:
Step 1: Consider the expression $(-6)^2$. This means that we take -6 and multiply it by itself:
$(-6) \times (-6) = 36$

Step 2: Consider the expression $-(6)^2$. Here, the square acts only on 6, not on the negative sign in front because of the absence of parentheses around -6:
$-(6 \times 6) = -36$

Therefore, the expression $(-6)^2$ correctly equals 36.

The correct choice that satisfies $36 =$ is $(-6)^2$.

To solve this problem, we'll need to carefully consider the placement of parentheses and the order of operations when dealing with negative numbers and exponentiation:

• Step 1: Examine the given expression $(-4)^2$. This indicates squaring the entire negative four, which results in $16$.
• Step 2: Now, examine $-(-4)^2$ paying close attention to the negative sign in front. $(-4)^2$ again results in $16$, but placing a minus before it changes its sign to $-16$.

Let's analyze each possible choice:

• For choice 1: The expression is $-(-4)^2$. Calculating $(-4)^2$ gives $16$, and applying the negative sign outside results in the expression value of $-16$.
• For choice 2: The expression is $(-4)^2$, calculated to give $16$.

Therefore, the expression that correctly equals $-16$ is choice 1, represented by $-(-4)^2$.

The correct answer is choice 1: $-(-4)^2$.

To solve this problem, we'll evaluate each expression step by step:

• Option a: $(-5)^2$
  This implies that we first compute $(-5) \times (-5)$ which equals $25$.

• Option b: $-(-5)^2$
  First, find $( -5)^2 = 25$. Then, apply the negative sign, resulting in $-25$.

• Option c: $-(5)^2$
  First, find $(5)^2 = 25$. Then, apply the negative sign, resulting in $-25$.

Options b and c both evaluate to $-25$. Therefore, the correct answer is option d: "Answers b and c".

To solve this problem, follow these steps:

• Step 1: Identify that the problem involves finding a mathematical expression that equals $-100$.
• Step 2: Consider how negative powers and negative bases interact with parentheses.
• Step 3: Evaluate each choice using the properties of powers.

Now, let's evaluate each choice:

Choice 1: $(-1)^{100}$
Computing: $(-1)^{100}$ equals $1$, since $100$ is even and any even power of $-1$ results in $1$. Therefore, this choice cannot be $-100$.

Choice 2: $(-10)^2$
Computing: $(-10)^2$ equals $100$, as squaring a negative number results in a positive number. Thus, this choice cannot be $-100$.

Choice 3: $-(-10)^2$
Computing: $-(-10)^2$ means first computing $(-10)^2$, which is $100$, and then applying the negative sign resulting in $-100$. Therefore, this correctly matches $-100$.

Choice 4: $1^{100}$
Computing: $1^{100}$ is $1$, as any number raised to any power remains itself when the base is $1$. Thus, it cannot equal $-100$.

Therefore, the correct choice is $-(-10)^2$, as it evaluates directly to $-100$.